On suficient conditions involving distances for hamiltonian properties in graphs
Let G be a 2-connected graph of order A set is essential if it is independent and contains two vertices at distance two apart. For = f1 2 3g we de…ne min max to be respectively the smallest, the second smallest and the largest value in f12 23 31g where = j() \ ( )j In this paper we show that the closure concept can be used to prove su¢cient conditions on hamiltonicity when distances are involved. As main results, we prove for instance that if either (i) each essential triple of satis…es the condition 2 () ¸ + or (ii) j() [ ()j + min f() ()g ¸ for all pairs of ( ) at distance two then its 0-dual closure is complete. By allowing classes of nonhamiltonian graphs we extend this result by one unit. A large number of new su¢cient conditions are derived. The proofs are short and all the results are sharp. Key words: Hamiltonian graph, closure, dual closure, neighborhood closure, essential sets.
